For \(r = 6, 7, . . . , 11\) we find an elliptic curve \(E/Q\) of rank at least \(r\) and the smallest conductor known, improving on the previous records by factors ranging from 1.0136 (for \(r = 6)\) to over 100 (for \(r = 10\) and \(r=11\)). We describe our search methods, and tabulate, for each \(r = 5, 6, . . . , 11\), the five curves of lowest conductor, and (except for \(r = 11)\) also the five of lowest absolute discriminant, that we found.